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On the Existence of Finite‐Dimensional Realizations for Nonlinear Forward Rate Models
Author(s) -
Björk Tomas,
Svensson Lars
Publication year - 2001
Publication title -
mathematical finance
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.98
H-Index - 81
eISSN - 1467-9965
pISSN - 0960-1627
DOI - 10.1111/1467-9965.00113
Subject(s) - forward rate , realization (probability) , affine transformation , short rate , mathematics , interest rate , nonlinear system , markov process , interest rate derivative , heath–jarrow–morton framework , short rate model , stochastic volatility , volatility (finance) , mathematical economics , yield curve , econometrics , pure mathematics , economics , finance , statistics , physics , quantum mechanics
We consider interest rate models of the Heath–Jarrow–Morton type, where the forward rates are driven by a multidimensional Wiener process, and where the volatility is allowed to be an arbitrary smooth functional of the present forward rate curve. Using ideas from differential geometry as well as from systems and control theory, we investigate when the forward rate process can be realized by a finite‐dimensional Markovian state space model, and we give general necessary and sufficient conditions, in terms of the volatility structure, for the existence of a finite‐dimensional realization. A number of concrete applications are given, and all previously known realization results (as far as existence is concerned) for Wiener driven models are included and extended. As a special case we give a general and easily applicable necessary and sufficient condition for when the induced short rate is a Markov process. In particular we give a short proof of a result by Jeffrey showing that the only forward rate models with short rate dependent volatility structures which generically possess a short rate realization are the affine ones. These models are thus the only generic short rate models from a forward rate point of view.